: What's the difference between Deus Ex Machina and coincidence? I've read 1, 2, 3 and of course, 4. Then I read 5 and 6 and even 7. I was blissfully ignorant about coincidence, DEM and plot-convenience;

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If a MC is lost in the woods and happen on a hermit living in a hut. The hermit gives directions to the MC to get out of the woods.

I don't even know if I'd call this a coincidence, assuming in your story there are hermits that live in the woods. Wondering around lost you would eventually be expected to find one of them and they would most likely give your character directions.
Compare that to:

MC is lost in the woods and comes across a pack stuffed with maps, compasses, and a solar charged GPS navigation device.

Totally unexpected, no reason for it to happen. DEM because you couldn't find some other way to resolve your character being lost. Not likely, the hermit or just letting your character find their way out of the woods would be easier, but you get the point.

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As an outsider looking in:
Deus Ex Machina
Is typically an out-of-nowhere event that effectively 'rescues the plot' that would have otherwise ended in a way the author didn't want. For example, the author writes a hyper-competent villain who successfully traps the heroes, and unable to convincingly write a way for them to escape, has magic space gods open a portal and free them. Classically the total opposite to a Chekov's gun.
Coincidence
A typically minor or less important event that usually occurs out of nowhere, which may make the reader go 'huh?' but won't necessarily seem like the author pulled it from nowhere, just it's missing explaination.
For example, coincidence that might be 'unexpected': the heroes stumble across the missing ring they were looking for in the middle of the forest, or the villains find some unlocked back door that's convenient to their plans for invasion.
An example of coincidence the reader might expect, would be the weather turning, either in favour or against one party or another, or someone winning a relatively low (but not impossible) probability event, such as a lottery or high-stakes game, or an action conducted based on intuition 'I had a feeling and picked this ring for you', 'that's my grandmother's ring!'
Summary
Coincidence does not necessarily benefit any party explicitly or push forward the plot, but can add in complications which later down the line impact the plot, where-as Deus Ex Machina fundamentally alters an otherwise logical outcome for a given plot point. In short: explicit intention.

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Go back to the origin of "Deus Ex Machina" - the term is literally ancient, referring to a god appearing in ancient (greek and roman) tragedy theatre plays to resolve the conflict.
When we use the term today, we generally refer to a turn of events with no visible relation to the story or characters, no foreshadowing, no causal link - just nothing.
A hut in a forest is not something unimaginable. The mentor of the character suddenly appearing from behind a tree to guide him would be.
The difference is that a coincidence can be unlikely, but is not impossible or unimaginable. It is within the realm of what could happen. Sure, the character got lucky stumbling upon that campsite with a map of the area, but people do go hiking and when they do they tend to carry maps.
The Deus Ex Machina is not unlikely, but implausible. Literally (Merriam-Webster) "provoking disbelief". A map falling out of the sky. With a red dot circling the protagonists current location. How exactly did that happen?
You can avoid most Deus Ex Machina impressions with proper writing. If the fact that a hermit is said to live in the woods is mentioned somewhere before, or if someone speaking about those woods a hundred pages earlier mentions that they are very sparsely populated, "you can find the occasional hunting lodge and the occasional loner living there, miles and miles from civilization", etc.

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The difference boils down to character arcs and agency.
To run with your "lost in the woods" example:
You could write a very satisfying story about being lost in the woods. You could describe how the threat to life and limb gives the main character a new appreciation of life, and a new understanding that all life is a struggle.
When they finally find the hermit, they know they survived only because they pushed on and overcame obstacles, even if there is an element of chance to this success.
Contrast this with the scenario where the main character gives up, and just lays down to wait for death. Now when the hermit arrives, the reader feels cheated.
If the end result is the same regardless of your character's actions, it might be Dues Ex Machina.

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The other answers cover the general idea of something being "too much", but don't really talk about the distinction between DEM and coincidence; this answer addresses that.
One way to think about it is that DEM requires the plot to be resolved through means not within the characters' control. This resolution may occur through random chance or happenstance, in which case it is also coincidence, but it could also be a logical extension of things that have already been established in the plot, in which case it's just DEM. You can also have coincidence which is not a DEM, where luck happens to be in the characters' favor, but it's what the characters do with that lucky break that resolves the plot.
An example of each:

Aliens are invading, everyone is dying, there is no hope, when all of a sudden the mortal enemies of the invading aliens show up and drive them off.

This is DEM but not necessarily coincidence. The resolution is dictated by something outside any of the characters' control, but could be made consistent with logic and additional storytelling: the invaders were trying to capture Earth so they could use its resources in their ongoing war, but the mortal enemies were tracking them and came to stop them as soon as they figured out what was happening.

Aliens are invading, everyone is dying, there is no hope, when all of a sudden the MC just happens to find an ancient relic that no one knew about, and it kills all the invaders.

This is DEM and coincidence because the resolution is abrupt and not really based on anything the characters did, and there isn't any particular reason the MC found the relic other than lucky chance.

Aliens are invading, everyone is dying, there is no hope. The MC is an expert in xenobiology, and has been working on understanding the aliens' DNA for years (they weren't always hostile). As luck would have it, during the invasion, an alien mothership crashes near MC's lab, which leads to the discovery of new information, that leads to a breakthrough, which leads to the development of a virus that kills off the invaders.

This is coincidence, but not DEM: the mothership crashing close by, and having the information that leads to the breakthrough are coincidence, but the ability to capitalize on those happenings and use them to resolve the plot is in the hands of the MC.

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Deus Ex Machina is when a chance event serves the purposes of the plot, leading to some outcome that is intended by the author or even sometimes desired by the audience or at any rate increases the dramatic effect of the whole story to the audience. Coincidence is when an event simply adds to the story-telling without significantly altering the plot.
As a result, coincidence is very rarely employed. In older story-telling, there was almost never any coincidence, only Deus Ex Machina. In modern story-telling, you do some times see coincidence, usually employed in order to add realism to the story.

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All Plots are Contrived
All stories are contrived. By this, I mean that they have been created by their authors. However, some writing is contrived well, and some poorly.
The key to contriving well is to keep the reader reading. There are a lot of guidelines on how to do this. You've linked several of them. The biggest, I've found, is to make sure that it appears as if your main character choices are important.
Let's take Disney's "Sorcerer's Apprentice" short, where the apprentice gets tired of mopping and casts a spell to animate the mops. From that decision comes the entire plot.
The Lord of the Rings starts with the decision made by Frodo to help Gandalf. It's a hard sell, because Hobbits are naturally homebodies, but in the end, it is his decision.
These decisions were made by the main character.
Now, I really do love Dave Weber's Out of the Night. It is an alien invasion movie with a human resistance. The main characters all make the decision to resist the aliens. However, after the half-way point, one of the resisters meets Dracula, and after that it's pretty much Deus Ex Vampire. From then on, Dracula drives the plot. Yes, I like the book, but it's poorly contrived.

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It all depends on how unlikely the event is, how vital the event is to the plot, and whether the reader/viewer has enough time to notice the holes.
For an example of bad plot-conveniences, consider the opening half-hour of Star Wars: A New Hope. Luke's reason for leaving Tatooine comes back to a crashlanded droid showed a transmission about someone called "Kenobi". We can discount the coincidences of the droids crashlanding on Tatooine and getting picked up by Jawas, because otherwise the Stormtroopers would never have come looking for them; and the Tantive IV wouldn't have been heading for Tatooine unless it was where Kenobi lived. So at least we don't have to justify why it's happening on that planet; and Jawas being Jawas, it's far more likely for the droids to be picked up by scavengers than that the droids stumble onto an outpost. But after that, the coincidences just stack up.

The Jawas don't visit anyone else before Owen (or at least no-one else wants the droids).
Owen happens to have had a droid fail on him the previous week, otherwise he wouldn't need one.
The other droid Owen buys from the Jawas fails immediately, otherwise he wouldn't need R2-D2.
Owen decides to buy both droids. (If he hadn't bought them, Luke wouldn't have met them.)
Luke has the necessary skills to enable R2-D2 to show the transmission.
Luke has transport.
Luke makes the connection to "Kenobi", a man he only knows by reputation, in spite of Episode 3 allegedly having Kenobi set up home on Tatooine to look after Luke.
Luke has the necessary curiosity (and boredom!) to go off on a highly dangerous trip to find Kenobi in spite of not knowing where he lives, instead of just telling R2-D2 and C-3PO to get on with their work.
Luke leaves before the Stormtroopers arrive, and doesn't (for example) spend the morning fixing the moisture collectors instead.
The Sand People don't just snipe him from a distance. (One of them considers it, but another one stops him for no obvious reason.)
For no reason at all, the Stormtroopers shoot Owen and Beru and torch the homestead. Had they asked questions first, Owen would happily have told them that he'd bought the droids, Luke had vanished with them, and they just needed to wait for him to get back. At worst, they would all have died there. At best, the Stormtroopers would just have confiscated the droids, and Luke would have lived and died on Tatooine without ever joining the Rebels.
Vader is unaware of the presence of both Luke and Kenobi on Tatooine, and Kenobi is similarly unaware of Vader's presence in orbit.

Not good, right? Of course if you push the plot forward fast enough then no-one really notices, so whether you get away with it or not will entirely depend on your writing style.
As @Andrey pointed out in a comment, all these take place in the first act. Some level of coincidence is usually required to set up a story where the little guy beats the big guy, because otherwise the story would just be "the Empire wins, game over". The issue for good writing is just how much coincidence your reader/viewer can stand before they say "wait, what?" @KevinWells has tried to justify this by "the Force exists, therefore it's destiny and not coincidence", but that really doesn't solve the problem because it takes agency away from the characters.
Conversely, Luke calling to Leia on Bespin in The Empire Strikes Back is a pure deus ex machina. We have no reason at that point to think this would do anything, we don't know they're brother and sister (and even Lucas hadn't written that yet, otherwise the kiss on Hoth wouldn't have happened!), we don't see him particularly using the Force in any way that Yoda might have shown him, and generally it just comes out of nowhere. Lucas has put Luke in an impossible position, and his only way out is to say "magic happens and the problem goes away".
Again as @Andrey says to make a very good distinction, deus ex machina happens at the end. It's not the setup to the story, it's the resolution. If the resolution is just "suddenly they're free/safe because magic happens", that's weak writing. Again, it takes agency away from the characters. There are times when characters should be left without agency (the point of most Godzilla-type monster movies is that the protagonists can't beat the monster), but that changes the nature of the story and you need to be careful not to leave your audience frustrated that you've invested all this time in characters who are snuffed for no particular reason.
It's probably not stretching too far to say that if you look at George Lucas's work, then you'll get good writing by doing the opposite of what he does. (As Harrison Ford famously said, "George, you can type this shit, but you can't say it.") On the other hand, Lucas's work has earned billions of dollars, so don't forget that good writing and popular, profitable writing are not always the same thing!

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There is no clear line. If everything else is going well, people won't notice that your coincidence is far fetched. The same plot resolution in another story might be considered "too much" to believe. But, let's try this:

A single action by one person (say, tripping and falling and hitting a lever that wasn't marked and that the person had no way of knowing about) completely eliminates all the problems (the multiple antagonists all fall into the open trapdoor, or all weapons stop working, etc) - too much
A series of actions, all in line with the character's, well, character, and all logical things for a person to do in that situation, leading to a success against strong odds - ok, even if the reader couldn't predict some of those actions.
A certain amount of unreliable narrator, where a character "steps out to chat with some locals" and then at the climax of the story the locals come riding over the hill with 50 of their friends and save our hero, as we're told that the chat was to plan this exact rescue -- it can work, especially if you've shown this character forming strong relationships and understanding the relevant group dynamics of the village or whatever

I think the key is the work you put in. If you get to the end and decide you want someone to push a Big Red Button and fix it, fine, but then go back and put the clues in. Have someone sit up late at night reading old notebooks and manuals. Or learning the ancient language -- and when they get to the room have them grin and say all that study time was worth it. Build up the parts of that character that make it believable they would be in a position to save everything just like that. Don't just say "Oh, did I not mention I was fluent in Ancient and studied all these control rooms for the last 20 years before you met me?" If there's magic in your world, give somebody a lucky talisman that makes even their trip-and-falls work for them, and show it working on a small scale long before the Big Moment.
You don't need to invent all that stuff before your DEM is written. You can go back and add it after you write the DEM. to turn the DEM back into a coincidence or even the only possible way this could have played out, which is what you want it to feel like.

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